The present invention relates to analysis of time varying data; and to analysis techniques based on fractals.
Frequently a time varying signal is required to be analyzed to extract characteristics of interest. For example, medical diagnosis often requires the analysis of time varying cardiac, respiratory, or brain signals in order to detect cardiac, pulmonary, or mental problems. In industrial control, electrical sensors produce signals in response to sensed parameters which occur over time in a manufacturing process and a control system responds to characteristics detected in the resultant signals. In general, most temporal processes are analyzed using Fourier Transform technique (frequency domain), chaos dynamics (position-velocity phase plane) and other complex mathematical techniques have been applied to signal analysis. A common drawback of these methods is that they are often complex, not easily amenable to analysis, and require some data pre-processing procedures, such as filtering, etc. Thus there remains a need for simple and practical methods for analyzing such time varying electrical signals.
Images and shapes within images which cannot be represented by Euclidean geometry have been analyzed by Fractal geometry. The term xe2x80x9cfractalsxe2x80x9d is derived from the Latin word fractus, the adjectival form of frangere, or xe2x80x9cto breakxe2x80x9d. Unlike conventional geometry, which deals with lines, triangles, circles, spheres and cones, fractal geometry is concerned with broken or xe2x80x9cfracturedxe2x80x9d shapes as so commonly found in nature. Such shapes simply do not vary in degree from conventional geometry (for example, clouds that are not spheres, trees that are not cones, and rivers that do not run straight). The concept of fractals and related applications has been established by Mandelbrot in his pioneering book xe2x80x9cThe Fractal Geometry of Naturexe2x80x9d (Mandelbrot, 1982).
Fractals have been used to describe objects and geometrical formations. Many structures exhibit an underlying geometric regularity, known as scale invariance or self-similarity, that is the tendency of natural forms to repeat themselves, as with the resemblance and shape between large branches of a tree and small twigs. If these objects are observed at different size scales, there is the same fundamental pattern that is encountered. This repetitive pattern defines the fractional, or fractal dimension of the object structure, Avnir. D., Editor, The Fractal Approach To Heterogeneous Chemistry: Surfaces, Colloids, Polymers, J. Wiley and Sons, New York 1992. Fractals exhibit the property of self similarity. The shapes do not have to be identical. Fractal systems accommodate structures within structures and occupy more space than ordered systems. Fractal analysis of antigen-antibody binding kinetics have been utilized to enhance the performance of biosensors (Sadana, Alarie and Vo-Dinh, 1995).
Chaos refers to a constrained type of unpredicted turbulent dynamics. Chaotic systems are characteristically very sensitive to initial conditions. Chaotic vibrations appear when some strong non-linearity occurs in the system. Chaotic vibrations have been observed in many physical systems, such as closed and open-flow fluid problems, chemical reactions, acoustic systems, cardiac oscillations, and earthquake dynamics. Skinner emphasizes that all fractal dimensional systems are chaotic, and that the data they generate will be aperiodic, complex, and apparently unpredictable (Skinner, xe2x80x9cLow-Dimensional Chaos in Biological Systemsxe2x80x9d, Bio/Technology, 12, 596-600 1994). The analysis of the dynamics of human biomedical or biological signals is an important area of investigation to help control and to be able to predict the onset of pathological conditions. Chaotic behavior is exhibited by the heart in electrocardiogram signals and by the brain in electroencephalogram (EEG) signals. It has been emphasized that the demonstration of chaotic behavior in humans opens out the possibility of rapid diagnosis and effective therapeutic control of conditions ranging from epilepsy to cardiac arrest. What is required is a practical method or procedure that utilizes or translates chaos and/or fractal theory concept into a simple and straightforward manner to help distinguish between normal and pathological behavior.
The composition of a physical material is often analyzed using spectroscopy, such as techniques based upon ultra-violet or infra-red absorption, and Raman scattering. Normal Raman spectroscopy relates to the scattering of usually monochromatic incident radiation by a gas, liquid or solid which produces a shift in frequency or wavelength. Upon irradiation of a molecule with light in biological applications, the incident radiation having a frequency n should produce scattered radiation, the most intense part of which has unchanged frequency (Rayleigh scattering). In addition, if the polarization of a molecule changes as it rotates or vibrates, there are spectral lines of much lesser intensity at frequencies nxc2x1nK, where nK is the molecular frequencies of rotation or vibration.
The results of Raman spectroscopy are frequently depicted in a two-dimensional image which is interpreted by human inspection. Thus it is desirable to provide a more efficient and repeatable analysis method for the Raman image.
The present invention involves a novel approach of using fractal techniques to analyze temporal events by conversion of representative temporal signals into spatial patterns. The proposed method of Fractal Analysis with Spacexe2x80x94Time (FAST) coordinate conversion is based on the concept that, when the temporal signal of a process is converted into a spatial pattern, the element of this spatial pattern can be characterized and analyzed by fractal geometry. This time-space conversion process is consistent with the concept that scale invariance has some parallel in chaos theory, which is generally used to analyze many temporal processes, such as atmospheric turbulence, cardiac rhythms, or mechanical operations. In fact, it has been indicated that chaotic behavior is present in quite a few biological processes that are occurring in the human body, and these give rise to the fractal structures that are prevalent in the body, Goldberg et al., xe2x80x9cChaos and Fractals in Human Physiologyxe2x80x9d, Scientific American, pp. 43-49, February 1990. In fact, relently fractal geometry has been referred to as the fourth dimension of life, West et al., American Science, vol. 284, p4, June 1999.
Fractal structures are often believed to be derived from the remnants of chaotic nonlinear dynamics. Quantitative tests for chaotic dynamics involve the analysis of several parameters, including the Lyapunov exponent and the fractal dimension in the phase space (Moon, Moon, F. Chaotic and Fractal Dynamics, Wiley, NY 1992). Although an interrelationship between chaos and fractal dynamics has been known, the concept of using fractal geometry directly to analyze a temporal process following space-time coordinate conversion, is not believed to have been suggested or used previously. The space-time conversion method is a direct, simple, and rapid process that does not require complex and time-consuming data conversion into the phase space.
Specifically characteristics of time varying data, such as an electrical signal, are analyzed by converting the data from a temporal domain into a spatial domain pattern. This may be accomplished by graphically plotting the time varying data as a two-dimensional image which then is scanned into a computer, or by electronically transforming the data into an image in the computer. Fractal analysis then is performed on the spatial domain pattern, thereby producing a fractal dimension DF which indicates the regularity or heterogeneity of the time varying data.
This technique has practical applicability in analyzing physiological data to diagnose disease in animals.